Numerical identification of a Robin coefficient in parabolic problems

Jin, Bangti; Lu, Xiliang

doi:10.1090/S0025-5718-2012-02559-2

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ISSN 1088-6842(online) ISSN 0025-5718(print)

Numerical identification of a Robin coefficient in parabolic problems

Authors: Bangti Jin and Xiliang Lu
Journal: Math. Comp. 81 (2012), 1369-1398
MSC (2010): Primary 65M30, 65M32, 65M12
Posted: January 11, 2012
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Abstract: This paper studies a regularization approach for an inverse problem of estimating a spatially-and-temporally dependent Robin coefficient arising in the analysis of convective heat transfer. The parameter-to-state map is analyzed, especially a differentiability result is established. A regularization approach is proposed, and the properties, e.g., existence and optimality system, of the functional are investigated. A finite element method is adopted for discretizing the continuous optimization problem, and the convergence of the finite element approximations as the mesh size and temporal step size tend to zero is established. Numerical results by the conjugate gradient method for one- and two-dimensional problems are presented.

[AF03] Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078 (2009e:46025)
[Ali94] Oleg M. Alifanov, Inverse heat transfer problems, Springer-Verlag, Berlin, 1994.
[BBSC85] James V. Beck, Ben Blackwell, and Charles R. St. Clair, Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, 1985.
[BCC08] Mourad Bellassoued, Jin Cheng, and Mourad Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math. Anal. Appl. 343 (2008), no. 1, 328–336. MR 2412131 (2009e:35299), http://dx.doi.org/10.1016/j.jmaa.2008.01.066
[BPS02] James H. Bramble, Joseph E. Pasciak, and Olaf Steinbach, On the stability of the 𝐿² projection in 𝐻¹(Ω), Math. Comp. 71 (2002), no. 237, 147–156 (electronic). MR 1862992 (2002h:65175), http://dx.doi.org/10.1090/S0025-5718-01-01314-X
[BS08] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954 (2008m:65001)
[BX91] James H. Bramble and Jinchao Xu, Some estimates for a weighted 𝐿² projection, Math. Comp. 56 (1991), no. 194, 463–476. MR 1066830 (91k:65140), http://dx.doi.org/10.1090/S0025-5718-1991-1066830-3
[CFJL04] S. Chaabane, I. Fellah, M. Jaoua, and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems 20 (2004), no. 1, 47–59. MR 2044605 (2005a:35276), http://dx.doi.org/10.1088/0266-5611/20/1/003
[CGH06] K. Chrysafinos, M. D. Gunzburger, and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE, J. Math. Anal. Appl. 323 (2006), no. 2, 891–912. MR 2260151 (2007h:49041), http://dx.doi.org/10.1016/j.jmaa.2005.10.053
[Cha99] S. Chantasiriwan, Inverse heat conduction problem of determining time-dependent heat transfer coefficient, Int. J. Heat Mass Transfer 42 (1999), no. 23, 4275-4285.
[Che98] Xu-Yan Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998), no. 4, 603–630. MR 1637972 (99h:35078), http://dx.doi.org/10.1007/s002080050202
[Cho99] Mourad Choulli, On the determination of an unknown boundary function in a parabolic equation, Inverse Problems 15 (1999), no. 3, 659–667. MR 1696926 (2001b:35291), http://dx.doi.org/10.1088/0266-5611/15/3/302
[CJ99] Slim Chaabane and Mohamed Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems 15 (1999), no. 6, 1425–1438. MR 1733209 (2000j:35270), http://dx.doi.org/10.1088/0266-5611/15/6/303
[CW08] Han-Taw Chen and Xin-Yi Wu, Investigation of heat transfer coefficient in two-dimensional transient inverse heat conduction problems using the hybrid inverse scheme, Internat. J. Numer. Methods Engrg. 73 (2008), no. 1, 107–122. MR 2378451 (2009b:65242), http://dx.doi.org/10.1002/nme.2059
[EG92] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
[Eva98] Lawrence C. Evans, Partial Differential Equations, AMS, Providence, R.I., 1998.
[Gr92] Konrad Gröger, 𝑊^{1,𝑝}-estimates of solutions to evolution equations corresponding to nonsmooth second order elliptic differential operators, Nonlinear Anal. 18 (1992), no. 6, 569–577. MR 1154481 (93a:35066), http://dx.doi.org/10.1016/0362-546X(92)90211-V
[HMRR10] D. Hömberg, C. Meyer, J. Rehberg, and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim. 48 (2009/10), no. 5, 3449–3481. MR 2599927 (2011b:49009), http://dx.doi.org/10.1137/080736259
[Jin07] Bangti Jin, Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Internat. J. Numer. Methods Engrg. 71 (2007), no. 4, 433–453. MR 2332815 (2008g:65146), http://dx.doi.org/10.1002/nme.1949
[JZ08] Bangti Jin and Jun Zou, Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comput. Phys. 227 (2008), no. 6, 3282–3306. MR 2392734 (2009c:65225), http://dx.doi.org/10.1016/j.jcp.2007.11.042
[JZ09] Bangti Jin and Jun Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48 (2009), no. 3, 1977–2002. MR 2516196 (2010h:65205), http://dx.doi.org/10.1137/070710846
[JZ10] Bangti Jin and Jun Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal. 30 (2010), no. 3, 677–701. MR 2670110 (2011f:65241), http://dx.doi.org/10.1093/imanum/drn066
[KJ01] Refahi Abou Khachfe and Yvon Jarny, Determination of heat sources and heat transfer coefficient for two-dimensional heat flow - numerical and experimental study, Int. J. Heat Mass Transfer 44 (2001), no. 7, 1309-1322.
[Kno98] Ian Knowles, A variational algorithm for electrical impedance tomography, Inverse Problems 14 (1998), no. 6, 1513–1525. MR 1662464 (99i:65127), http://dx.doi.org/10.1088/0266-5611/14/6/010
[Kre00] F Kreith (ed.), The CRC Handbook of Thermal Engineering, CRC, Bota Raton, 2000.
[KS80] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. MR 567696 (81g:49013)
[LU68] Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York, 1968. MR 0244627 (39 #5941)
[LW93] S. Lenhart and D. G. Wilson, Optimal control of a heat transfer problem with convective boundary condition, J. Optim. Theory Appl. 79 (1993), no. 3, 581–597. MR 1255288 (94j:49005), http://dx.doi.org/10.1007/BF00940560
[Miz58] Sigeru Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 31 (1958), 219–239 (French). MR 0106347 (21 #5081)
[NH94] Dinh Nho Hào, A noncharacteristic Cauchy problem for linear parabolic equations and related inverse problems. I. Solvability, Inverse Problems 10 (1994), no. 2, 295–315. MR 1269009 (95b:35228)
[NHR97] Dinh Nho Hào and H.-J. Reinhardt, On a sideways parabolic equation, Inverse Problems 13 (1997), no. 2, 297–309. MR 1445920 (97m:35273), http://dx.doi.org/10.1088/0266-5611/13/2/007
[OB88] A. M. Osman and J. V. Beck, Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients, J. Thermophys. 3 (1988), no. 2, 146-152.
[OIL08] T. T. M. Onyango, D. B. Ingham, and D. Lesnic, Reconstruction of heat transfer coefficients using the boundary element method, Comput. Math. Appl. 56 (2008), no. 1, 114–126. MR 2427690 (2009f:80017), http://dx.doi.org/10.1016/j.camwa.2007.11.038
[SH04] Jian Su and Geoffrey Hewitt, Inverse heat conduction problem of estimating time-varying heat transfer coefficient, Numer. Heat Transfer Part A 45 (2004), no. 8, 777-789.
[SLO10] M. Slodička, D. Lesnic, and T. T. M. Onyango, Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem, Inverse Probl. Sci. Eng. 18 (2010), no. 1, 65–81. MR 2598682 (2011a:65304), http://dx.doi.org/10.1080/17415970903234273
[SV02] Marián Slodička and Roger Van Keer, Determination of a Robin coefficient in semilinear parabolic problems by means of boundary measurements, Inverse Problems 18 (2002), no. 1, 139–152. MR 1893587 (2003d:35271), http://dx.doi.org/10.1088/0266-5611/18/1/310
[TA77] Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR 0455365 (56 #13604)
[Whi88] F. M. White, Heat and mass transfer, Addison-Wesley, Reading, 1988.
[XZ05] Jianli Xie and Jun Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal. 43 (2005), no. 4, 1504–1535. MR 2182138 (2006e:35344), http://dx.doi.org/10.1137/030602551
[YYW09] Fenglian Yang, Liang Yan, and Ting Wei, The identification of a Robin coefficient by a conjugate gradient method, Internat. J. Numer. Methods Engrg. 78 (2009), no. 7, 800–816. MR 2514342 (2010b:65194), http://dx.doi.org/10.1002/nme.2507
[ZK79] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979), no. 1, 49–62. MR 526427 (82a:90153), http://dx.doi.org/10.1007/BF01442543

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